Upper bound for the number of zeros of a meromorphic function outside a vertical line and applications. (Majoration du nombre de zéros d’une fonction méromorphe en dehors d’une droite verticale et applications.) (French, English) Zbl 1203.11059

Let \(a\in{\mathbb R}\), \(h(s)\) a meromorphic function in \({\mathbb C}\), real on \({\mathbb R}\), with only finitely many poles, finitely many zeros in the half-space \(\sigma>a\), holomorphic and non-zero on the critical line \(\sigma=a\). Let \(f(s)=f^{\pm}(s)=h(s)\pm h(2a-s)\). Suppose that the function \(F(s)=\frac{h(2a-s)}{h(s)}\) satisfies
(i) for every \(\eta>0\) there exists \(\sigma_0=\sigma_0(\eta)>a\) such that \(|F(s)|<\eta\) if \(\sigma\geq \sigma_0\) (\(s=\sigma+i\tau\));
(ii) for every \(\varepsilon>0\) and \(\sigma_0>a\) there exists a sequence \((T_n)_n \) such that \(\lim_{n\to\infty}T_n=\infty \) and \(|F(s)|<e^{\varepsilon |s|}\) for \(a\leq \sigma\leq \sigma_0,|\tau|=T_n,n\geq 1\).
The author proves that all zeros of \(f(s)\) up to a finite number lie on the line \(\sigma=a\) and are simple. He studies translations of the Riemann zeta-function and \(L\)-functions, and integrals of Eisenstein series, among others.


11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses


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